Fraction becoming whole deal for students

Joy Awumi, right, uses an iPad and plastic shapes to learn about fractions at the Sylvan Learning Center in Worcester with teacher Julie Camfield.

Helping students understand that fractions are numbers, and not "something different" using tricks and formulas, is a primary aim of the Common Core State Standards being implemented throughout Massachusetts public schools, advocates of the initiative say.

The Common Core introduces fractions in third grade. Previously, at least in many math programs, they hadn't come up until fourth grade, when students were "slammed" with various concepts around fractions, said Linda M. Gojak, director of the Center of Mathematics and Science Education Teaching and Technology in Ohio.

Much of the former ideal was on display July 30 at the Sylvan Learning Center on Pullman Street in Worcester, which is conducting classes to help students understand the Common Core Standards.

During one session, Joy Awumi of Worcester, who is headed to third grade, used an iPad and shapes during her introduction to basic fractions.

Center director Julie Camfield, who said she remembered doing fractions in middle school, helped Joy divide regions into subregions that were equivalent.

The tandem examined circles, triangles and rectangles that were split into halves and thirds, with the goal of getting Joy to understand that two or three of each part made a whole.

While detractors of the standards argue that key math topics are missing or appear in the wrong grade, Ms. Camfield said early elementary grades were the perfect time to introduce fractions and other important concepts, which can help a child master future math skills such as algebra.

"We see lots of high school students come to us for help with their math classes, and our assessments often reveal a weak foundation in fractions," she said.

Ms. Camfield insisted that the day's lesson for Joy was something a second- or third-grader could wrap their heads around.

"This is where we start," she said.

By the end of the session, Joy had managed to figure out how to divide fractions through the shapes, but Ms. Camfield said she needed to continue to work with her on how to talk about the concept in terms of one whole.

The next time Joy comes in, she is to move on to independent practice.

Laureen Cipolla, science and mathematics administrator at Leominster schools, said that while the new Massachusetts frameworks incorporating the Common Core noted that mastery of fractional understanding for students had been accelerated, at the same time, the progression of learning is well-articulated.

For example, fifth-grade students divide with fractions, a standard previously begun in sixth grade.

This progression mirrors the progression of student understanding in whole numbers, she said.

Ms. Cipolla said a major stumbling block regarding teaching and learning fractions lies in the misconception that fractions are somehow different from whole numbers.

"Our instruction needs to build on the students' understanding of whole numbers and make connections to their understandings about fractions," she said.

Leominster and Fitchburg school districts recently teamed up to support students' mastery of the new standards. The two districts completed a one-week "Lab Class" with more than 30 teachers and 20 students focusing on the progression of fractional understanding, Ms. Cipolla said.

By the end of the week, the students had engaged in challenging math and science learning, she said. Teachers had opportunities to examine their own understanding of fractions, to observe students thinking about fractions, and to practice new instruction and assessment under the guidance of content and instruction experts, Ms. Cipolla said.

Ms. Gojak, who directs the center in Ohio, said she supported the Common Core's approach to understanding of fractions as a unit piece, similar to comparing inches and centimeters.

"What the Common Core does is develop those fundamental underpinnings and structure of fractions in Grade 3, and then they start to build on them in Grades 4 and 5," she said.

Ms. Gojak also said she appreciated that third-graders work specifically with denominators of 2, 3, 4, 6 and 8, numbers with which they will likely have had everyday experiences: Cutting something in half or thirds, or a four-paned window.

Ms. Gojak acknowledged that this earlier and deeper conceptual approach was hard to understand through a reading of the standards, which are written in complex math language.

Ms. Gojak said she also supported the Common Core's emphasis on thinking about the size of fractions. It gives students the concept that 5/6 is close to a whole, while 3/5 is close to a half.

Ms. Gojack said she had worked closely with both bright students and those who struggled, and both groups' understanding of fractions were fragile, even in fifth and sixth grade.

"This idea (is) of giving kids these relationships, and seeing things in context that they can understand before they get into more abstract ways of thinking about fractions," she said. "Not just showing them a bunch of rules."

Some studies have found that children below third grade understand some concepts of fractions, said Nancy C. Jordan of the Center on Improving Mathematics Instruction for Students with Mathematics Difficulties, which is administered at the University of Delaware.

It's important to get children thinking about fractions on a number line between 0 and 1, more than thinking of fractional numbers on a basic level, said Ms. Jordan, whose center focuses on improving math instruction for elementary and middle school children who have problems with math concepts, specifically fractions. It received a $10 million grant in 2010 from the Institute of Education Sciences.